Labrune said he would talk to players on Friday in a meeting scheduled even before the 3-2 extra-time defeat to the third-tier minnows.
"Everyone has to face up to his responsibilities,” Labrune told AFP. “There's one thing that we can't comprise on, that's respect for the Marseille jersey."
The defeat is the latest setback for the French giants who, despite qualifying for the quarter-finals of the Champions League with victory over Internazionale, are struggling in Ligue 1.
Relegation-threatened Dijon dented Marseille's hopes of European football next season with a shock 2-1 win on Saturday, consigning them to a fifth straight defeat and leaving them ninth in the league.
"The priority now is to show another side in the championship and to give our all to finish as high as possible," he said.
Labrune added: "The only consolation after this French Cup defeat was that the fixture list will be less hectic.”
Meanwhile, Marseille have asked for their March 31 league match against Montpellier to be put back to April 3 as they prepare for their Champions League quarter-final against Bayern Munich.
French league officials are expected to give their decision on the request on Thursday.